Skip to content
ToolUlkaa Tool
All tools

Mathematics

Percentage CalculatorFraction CalculatorEquation SolverTensorMatrix CalculatorUnit ConverterNumber Base ConverterDerivative CalculatorIntegral CalculatorLimit CalculatorAlgebra CalculatorQuadratic Formula CalculatorFactoring CalculatorLogarithm CalculatorSquare Root CalculatorLong Division CalculatorSignificant Figures CalculatorLCM and GCF CalculatorPrime Factorization CalculatorTriangle CalculatorPythagorean Theorem CalculatorSlope CalculatorArea CalculatorVolume CalculatorScientific CalculatorGraphing Calculator

Categories

AcademicTextMathematicsStatisticsMachine LearningAIData ScienceFile
All tools
Mathematics

Completing the Square Calculator

Enter a, b and c to see the equation rewritten as a perfect square, step by step.

Show
Enter
of x squared
of x
constant
Try:

Completing the square

y=2(x+2)2−5y = 2\left(x + 2\right)^2 - 5y=2(x+2)2−5

Two real roots, x = (-4 + √10)/2 and x = (-4 - √10)/2.

Forms of the same parabola

Standard formy=2x2+8x+3y = 2x^{2} + 8x + 3y=2x2+8x+3
Factored formy=2(x−(−4+102))(x−(−4−102))y = 2\left(x - \left(\frac{-4 + \sqrt{10}}{2}\right)\right)\left(x - \left(\frac{-4 - \sqrt{10}}{2}\right)\right)y=2(x−(2−4+10​​))(x−(2−4−10​​))
Vertex formy=2(x+2)2−5y = 2\left(x + 2\right)^2 - 5y=2(x+2)2−5
Vertex(−2, −5)\left(-2,\ -5\right)(−2, −5)
Axis of symmetryx=−2x = -2x=−2
y-intercept(0, 3)\left(0,\ 3\right)(0, 3)
DiscriminantΔ=40\Delta = 40Δ=40

The roots are irrational, so the factors contain square roots. The parabola opens upward, so the vertex is a minimum.

Parabola

  • y
Table of values
Graph of the parabola with its roots and vertex marked: sampled values across the visible range
xy
-5.794723.8
-5.035813.432
-4.27685.368
-3.5179-0.392
-2.7589-3.848
-2-5
-1.2411-3.848
-0.48211-0.392
0.276845.368
1.035813.432
1.794723.8

Completing the square, step by step

  1. 1Divide by a
    2x2+8x+3=02x^{2} + 8x + 3 = 02x2+8x+3=0
    ↓
    x2+4x=−32x^2 + 4x = -\frac{3}{2}x2+4x=−23​

    Divide every term by a so the x squared term has coefficient 1.

  2. 2Add the square of half the x coefficient
    (b2a)2\left(\frac{b}{2a}\right)^2(2ab​)2
    ↓
    (2)2=4\left(2\right)^2 = 4(2)2=4

    Add this to both sides.

  3. 3Write the left side as a square
    x2+4x+4=52x^2 + 4x + 4 = \frac{5}{2}x2+4x+4=25​
    ↓
    (x+2)2=52\left(x + 2\right)^2 = \frac{5}{2}(x+2)2=25​

    The right side equals the discriminant over 4a squared: 40 / 16.

  4. 4Take the square root of both sides
    x+2=±52x + 2 = \pm\sqrt{\frac{5}{2}}x+2=±25​​
    ↓
    x+2=±102x + 2 = \pm \frac{\sqrt{10}}{2}x+2=±210​​
  5. 5Vertex form
    y=2x2+8x+3y = 2x^{2} + 8x + 3y=2x2+8x+3
    ↓
    y=2(x+2)2−5y = 2\left(x + 2\right)^2 - 5y=2(x+2)2−5

    The same completing-the-square work, kept as an expression, gives the vertex.

Steps

  1. 1Identify the coefficients
    ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0
    ↓
    2x2+8x+3=02x^{2} + 8x + 3 = 02x2+8x+3=0

    a = 2, b = 8, c = 3

  2. 2Discriminant
    Δ=b2−4ac\Delta = b^2 - 4acΔ=b2−4ac
    ↓
    Δ=82−4(2)3=40\Delta = 8^2 - 4(2)3 = 40Δ=82−4(2)3=40

    Positive: two different real roots.

  3. 3Quadratic formula
    x=−b±Δ2ax = \frac{-b \pm \sqrt{\Delta}}{2a}x=2a−b±Δ​​
    ↓
    x=−8±404x = \frac{-8 \pm \sqrt{40}}{4}x=4−8±40​​
  4. 4Simplify the square root
    40\sqrt{40}40​
    ↓
    2102\sqrt{10}210​

    Take out the largest square factor.

  5. 5The two roots
    x=−8±2104x = \frac{-8 \pm 2\sqrt{10}}{4}x=4−8±210​​
    ↓
    x1=−4+102,x2=−4−102x_1 = \frac{-4 + \sqrt{10}}{2}, \quad x_2 = \frac{-4 - \sqrt{10}}{2}x1​=2−4+10​​,x2​=2−4−10​​
  1. Tools
  2. Mathematics
  3. Quadratic Formula Calculator
  4. Completing the Square Calculator

About Completing the Square Calculator

This completing the square calculator solves ax² + bx + c = 0 by turning the left side into a perfect square, and writes every step so you can follow or check your own working. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.

The method is: divide by a, move the constant across, add the square of half the x coefficient to both sides, write the left side as a square, then take square roots. The page opens on 2x² + 8x + 3, where dividing by 2 leaves fractions and the square to add is 4.

The finished answer matches the quadratic formula: (-4 + √10)/2 and (-4 - √10)/2. The same working yields the vertex form of the parabola.

How to use Completing the Square Calculator

  1. 1Choose what to show: Roots, Discriminant, Vertex or Completing the square.
  2. 2Enter a, b and c in their boxes, or switch to Equation and type something like 2x^2 - 7x + 3 = 0.
  3. 3Read the headline answer. It updates as you type, with no calculate button.
  4. 4Open the steps underneath to follow the substitution into the quadratic formula, or the completing-the-square route.
  5. 5Use the table for the factored form, vertex form, axis of symmetry and y-intercept, and the graph to see where the parabola crosses the x axis.

Features

  • Exact roots: rational, simplified radical or complex, never only a rounded decimal.
  • Discriminant with a reading: two real roots, one repeated root or a complex pair.
  • Vertex, axis of symmetry, y-intercept and whether the parabola opens up or down.
  • Factored form over the rationals when it exists, with the complex factorisation otherwise.
  • Completing the square written out, ending in vertex form.
  • Quadratic formula steps in KaTeX, with a copy button for the LaTeX.
  • Interactive plot with roots and vertex marked, zoom, pan and PNG export.
  • Accepts integers, decimals, fractions and a full equation.

Completing the square

Add (b / 2a)² to both sides after dividing by a.

x² + (b/a)x = -c/a
(x + b/(2a))² = (b² - 4ac) / (4a²)

Worked examples

  • x² + 6x + 2 = 0 → gives x = -3 + √7 and x = -3 - √7

    Half of 6 is 3 and 3² = 9, so (x + 3)² = 7.

  • 2x² + 8x + 3 = 0 → gives x = (-4 + √10)/2 and x = (-4 - √10)/2

    Dividing by 2 gives x² + 4x = -3/2, then (x + 2)² = 5/2.

Privacy: how it works

  • Everything is calculated by code running in your browser tab. The coefficients you enter are not sent to Ulkaa or to any other server, and there is nothing to sign in to.

Tips

  • Check an answer by substituting a root back into the equation: the left side should come out as 0.
  • If the roots look awkward, look at the discriminant first. A perfect square means rational roots and a clean factorisation.
  • The sum of the roots is -b/a and their product is c/a, which is a quick check on any pair of roots.

Frequently asked questions

Is it really free?

Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.

Do I need an account?

No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.

Does it run in my browser?

Yes. Coefficients are processed by code running in your browser tab, on your own device.

Is my data uploaded?

No. Coefficients are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.

What are the steps of completing the square?

Divide by a, move c/a to the right, add (b/2a)² to both sides, write the left as (x + b/2a)², then take the square root of both sides and solve for x.

When is completing the square better than the formula?

When you need vertex form, or when a is 1 and b is even, where the arithmetic is short. For messy coefficients the formula is usually quicker.

More Quadratic Formula Calculator pages

  • Quadratic Formula CalculatorEnter a, b and c, or type the equation, and get the roots with every step, the vertex and a graph of the parabola.
  • Discriminant CalculatorEnter a, b and c to get the discriminant and what it says about the roots.
  • Vertex Form CalculatorEnter a, b and c to get the vertex, the axis of symmetry and the vertex form.
  • Quadratic Equation With Complex RootsSolve a quadratic with no real roots and get the complex pair, with the working.

Ulkaa Apps

  • Ulkaa
  • Calendar
  • Task
  • Note
  • Disk
  • Call
  • Contact
  • Mail
  • Finance
  • Health
  • Office
  • Work
  • Workspace
  • Halum
  • Jaal
  • Islam
  • Game
  • Weather
  • Campus
  • News
  • Developer
  • Help
  • Account
About·Privacy·Terms·© 2026 Ulkaa

What is Ulkaa Tool?

Ulkaa Tool is a free collection of fast, privacy-friendly online tools from Ulkaa Network. Calculate your GPA and CGPA, edit rich text, count words and characters, replace text and more — all in your browser.

Every tool runs entirely in your browser — no sign-up, and your data never leaves your device.

Academic

  • GPA Calculator
  • CGPA Calculator
  • Citation Generator
  • Quiz Maker

Text

  • Rich Text Editor
  • Word Counter
  • Character Counter
  • Text Replacement

Ulkaa

  • Ulkaa
  • Ulkaa News
  • Ulkaa Campus
  • Ulkaa Tool
  • Ulkaa Game
  • Ulkaa Weather
  • Ulkaa Islam

Company

  • About Ulkaa
  • Careers
  • Support
  • Help
  • Privacy Policy
  • Terms of Service
  • Disclaimer
  • Sitemap
© 2015-26, Ulkaa. All rights reserved.
Facebook PageLinkedIn CompanyX (Twitter) ProfileInstagram ProfileGitHub Account